The fifty-six-dimensional module for \(E_ 7\). I: A four form for \(E_ 7\)
With the aim of classifying maximal subgroups in later work, the author develops a very explicit description of the Chevalley group \(E\) of type \(E_7\) over a field \(K\) of characteristic different from two. He starts with constructing a 4-form \(F\) on the 56 dimensional space \(M= (\bigwedge^2 V) \oplus (\bigwedge^2 V^*)\), where \(V= K^8\) and \(V^*\) is its \(K\)-linear dual. The automorphism group of the form \(F\) is \(E\) and \(F\) generates the ring of invariants \(K[M]^E\). A special collection of 1-spaces in \(M\) is said to consist of singular points. Enriching it with `lines', he gets a Lie incidence geometry whose automorphism group is \(E\) again. Other orbits of 1-spaces is \(M\) are also determined, with their stabilizers.
scientific article; zbMATH DE number 763949